Classification theorem for smooth social choice on a manifold

A classification theorem for voting rules on a smooth choice space W of dimension w is presented. It is shown that, for any non-collegial voting rule, σ, there exist integers v ^*(σ), w ^*(σ) (with v ^*(σ)<w ^*(σ)) such that

1. structurally stable σ-voting cycles may always be constructed when w ? v ^*(σ) + 1
2. a structurally stable σ-core (or voting equilibrium) may be constructed when w ? v ^*(σ) ? 1

Finally, it is shown that for an anonymous q-rule, a structurally stable core exists in dimension \(\frac{{n - 2}}{{n - q}}\) , where n is the cardinality of the society.